MinT - Best Known (39, 39+35, s)-Nets in Base 3

Best Known (39, 39+35, s)-Nets in Base 3

(39, 39+35, 44)-Net over F₃ — Constructive and digital

Digital (39, 74, 44)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 24, 16)-net over F₃, using
  - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (15, 50, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]

(39, 39+35, 53)-Net over F₃ — Digital

Digital (39, 74, 53)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(39, 39+35, 385)-Net in Base 3 — Upper bound on s

There is no (39, 74, 386)-net in base 3, because

1 times m-reduction [i] would yield (39, 73, 386)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 69562 044471 191928 347903 275443 298149 > 3⁷³ [i]